A295761 G.f. A(x) satisfies: A(x - A(x^2)) = x + 2*A(x^2).

1, 3, 6, 24, 96, 396, 1728, 7839, 36438, 172680, 831624, 4058202, 20021268, 99697188, 500429016, 2529375300, 12862429920, 65760468840, 337817930184, 1742850773154, 9026374329108, 46912014922392, 244588357460448, 1278937818954306, 6705339839722404, 35241796466506908, 185643541655678184, 979972436105339856, 5183169679909147200, 27464173024052341200

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1030

Formula

G.f. A(x) satisfies:

(1) A(x) = x + 3 * A( (2*x + A(x))^2/9 ).

(2) A(x) = -2*x + 3 * Series_Reversion( x - A(x^2) ).

(3) x = A( -x/2 + 3/2 * Series_Reversion( x + 2*A(x^2) ) ).

(4) A(x - A(x^2)) = x + 2*A(x^2).

Example

G.f.: A(x) = x + 3*x^2 + 6*x^3 + 24*x^4 + 96*x^5 + 396*x^6 + 1728*x^7 + 7839*x^8 + 36438*x^9 + 172680*x^10 + 831624*x^11 + 4058202*x^12 +...
such that  A(x - A(x^2))  =  x + 2*A(x^2).
RELATED SERIES.
A(x - A(x^2)) = x + 2*x^2 + 6*x^4 + 12*x^6 + 48*x^8 + 192*x^10 + 792*x^12 + 3456*x^14 + 15678*x^16 + 72876*x^18 + 345360*x^20 + 1663248*x^22 + 8116404*x^24 + 40042536*x^26 + 199394376*x^28 + 1000858032*x^30 + 5058750600*x^32 +...
which equals x + 2*A(x^2).
Series_Reversion( x - A(x^2) ) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 132*x^6 + 576*x^7 + 2613*x^8 + 12146*x^9 + 57560*x^10 + 277208*x^11 + 1352734*x^12 +...
which equals (A(x) + 2*x)/3.
A( (2*x + A(x))^2/9 ) = x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 132*x^6 + 576*x^7 + 2613*x^8 + 12146*x^9 + 57560*x^10 + 277208*x^11 + 1352734*x^12 +...
which equals (A(x) - x)/3.

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = -2*x + 3*serreverse(x - subst(A, x, x^2) +x^2*O(x^n)) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A141200, A295760, A295762.

Sequence in context: A327643 A296215 A152761 * A262349 A109155 A338112

Adjacent sequences: A295758 A295759 A295760 * A295762 A295763 A295764

Keyword

nonn

Author

Paul D. Hanna, Dec 03 2017

Status

approved